Hopf theorem

Short description: Topological degree is the only homotopy invariant of continuous maps to spheres

The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres.

Formal statement

Let M be an n-dimensional compact connected oriented manifold and $S^n$ the n-sphere and $f,g:M\to S^n$ be continuous. Then $\mathrm{deg}(f)=\mathrm{deg}(g)$ if and only if f and g are homotopic.

References

• Milnor, John W. (1997). Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 978-0-691-04833-8.